Finite Group Databasebeta

← #31900537

Subgroups — #31900537

Order 18176.

454 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589168C211yesmaximal, characteristic
2#22586899C212yesmaximal
3#22586198C212yesmaximal, complemented
4#22586173C211yesmaximal, characteristic, complemented
5C71×#2862C211yesmaximal, characteristic, complemented
6#17986549C2211yescharacteristic
7#17986052C2212yes—
8#17986023C2211yescharacteristic
9C2×#14556056—21no—
10C2×#14556056—21no—
11#17985732—24no—
12C2×C4⋊Q8×C71—21no—
13#391×C71—24no—
14#382×C71—21no—
15#382×C71C2212yes—
16C43×C71C2211yescharacteristic
17C2×#11765865—21no—
18C2×#11765849—21no—
19#14556128Q811yescharacteristic, complemented
20#14556125Q811yescharacteristic, complemented
21#14556116—21no—
22#14556102—22no—
23C4×D284—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556056—42no—
27#14556056—42no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—22no—
30C22×Q8×C71—22no—
31C4⋊Q8×C71—42no—
32C4⋊Q8×C71—42no—
33C2×C4⋊C4×C71—22no—
34C2×C4⋊C4×C71—22no—
35C2×C42×C71—22no—
36C2×C42×C71D411yescharacteristic
37C2×C42×C71D411yescharacteristic
38C2×C42×C71C2311yescharacteristic
39#95×C71—24no—
40#95×C71—22no—
41#95×C71D414yes—
42#11765865—41no—
43#11765865—41no—
44#11765865—41no—
45#11765865—41no—
46C2×Dic142—21no—
47C2×D284—21no—
48C2×D284—22no—
49C2×C22⋊D71—41no—
50C2×C4×D71—41no—
51#11765849—41no—
52#11765849—41no—
53C2×C71⋊C8—42no—
54C2×C71⋊C8—41no—
55C4⋊Dic71—22no—
56(C2×D71)⋊C4—42no—
57C2×Q8×C71—44no—
58C2×Q8×C71—48no—
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—24no—
62C22×C4×C71C2×D412yes—
63C22×C4×C71C2×D411yescharacteristic
64C4⋊C4×C71—44no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—42no—
67C42×C71—21no—
68C42×C71—21no—
69C42×C71—44no—
70C42×C71D812yes—
71C42×C71SD1612yes—
72C42×C71C2×Q811yescharacteristic
73C42×C71#4111yescharacteristic
74C42×C71#4112yes—
75C23×C71#12011yescharacteristic
76C22×D71—41no—
77Q8×C71—88no—
78C2×C4×C71—24no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—48no—
84C2×C4×C71—44no—
85C2×C4×C71C22⋊Q812yes—
86C2×C4×C71C22⋊Q811yescharacteristic
87C2×C4×C71#12412yes—
88C2×C4×C71#12411yescharacteristic
89C2×C4×C71C2×D811yescharacteristic
90C2×C4×C71C2×SD1611yescharacteristic
91C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C71#13711yescharacteristic
93C22⋊D71—81no—
94C22⋊D71—81no—
95C2×Dic71—41no—
96D284—41no—
97D284—41no—
98D284—82no—
99C4×D71—81no—
100C4×D71—81no—
101Dic142—41no—
102Dic142—41no—
103C71⋊C8—82no—
104C71⋊C8—81no—
105C22×C71—22no—
106C22×C71#39311yescharacteristic
107C22×C71#44711yescharacteristic
108C22×C71#44811yescharacteristic
109C2×D71—81no—
110C2×D71—82no—
111C4×C71—24no—
112C4×C71—22no—
113C4×C71—22no—
114C4×C71—42no—
115C4×C71—84no—
116C4×C71#47411yescharacteristic
117C4×C71#47611yescharacteristic
118C4×C71#48611yescharacteristic
119C4×C71#48911yescharacteristic
120Dic71—81no—
121#10362—711nomaximal, Sylow
122C2×C71—21no—
123C2×C71—21no—
124C2×C71#202311yescharacteristic
125C2×C71#203112yes—
126D71—161no—
127#2862D7111yescharacteristic, complemented
128#1962—711no—
129#1517—712no—
130#1286—712no—
131#1276—711no—
132C71#1036211yescharacteristic, complemented, Sylow
133#532—711no—
134C2×C4⋊Q8—21no—
135#431—712no—
136#423—711no—
137C2×#104—1421no—
138C2×#104—1421no—
139#391—24no—
140#382—21no—
141#382C2×D7112yes—
142C43C2×D7111yescharacteristic
143#327—1424no—
144C2×#41—1421no—
145C22×Q8—22no—
146C2×M16—1421no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—711no—
150#127—711no—
151#124—1421no—
152#121—1422no—
153C4×D4—1422no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C42—22no—
157C2×C42C22⋊D7111yescharacteristic
158C2×C42C22⋊D7111yescharacteristic
159C2×C42C22×D7111yescharacteristic
160C4⋊C8—1422no—
161C4⋊C8—1422no—
162#104—2842no—
163#104—2842no—
164C22⋊C8—1422no—
165C4×C8—1422no—
166#95—24no—
167#95—22no—
168#95C22⋊D7114yes—
169#41—2841no—
170#41—2841no—
171#41—2841no—
172#41—2841no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—1421no—
176C2×D4—2841no—
177C2×D4—1421no—
178C2×D4—1422no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—2841no—
183C22×C4C2×C22⋊D7112yes—
184C22×C4C2×C22⋊D7111yescharacteristic
185M16—2841no—
186M16—2841no—
187C2×C8—2842no—
188C2×C8—2841no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—1422no—
193C22⋊C4—2842no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1176585012yes—
198C42#1176585212yes—
199C42Q8×D7111yescharacteristic
200C42#1176586611yescharacteristic
201C42#1176586612yes—
202C23—2841no—
203C23#1455609711yescharacteristic
204Q8—88no—
205Q8—2841no—
206Q8—2841no—
207D4—2841no—
208D4—2841no—
209D4—5682no—
210D4—5681no—
211D4—5681no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—2841no—
220C2×C4—5681no—
221C2×C4—5681no—
222C2×C4#1455610412yes—
223C2×C4#1455610411yescharacteristic
224C2×C4#1455611712yes—
225C2×C4#1455611711yescharacteristic
226C2×C4C2×#1176585011yescharacteristic
227C2×C4C2×#1176585211yescharacteristic
228C2×C4#1455615411yescharacteristic
229C2×C4#1455615911yescharacteristic
230C8—5682no—
231C8—5681no—
232C22—22no—
233C22—5681no—
234C22—5682no—
235C22#1798591011yescharacteristic
236C22#1798613311yescharacteristic
237C22#1798613811yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—84no—
243C4—5681no—
244C4#1798627611yescharacteristic
245C4#1798628611yescharacteristic
246C4#1798634711yescharacteristic
247C4#1798636111yescharacteristic
248C2—21no—
249C2—21no—
250C2—11361no—
251C2#2258947311yescharacteristic
252C2#2258950912yes—
subgroupG/H
centerC22#17985910
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2862C2
socleC22×C71#393
prime radical#17986549C22
hypercenter#2862D71
hypocenterC71#10362
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22586198C2semidirect
4. #22586173C2semidirect
5. C71×#2862C2semidirect
19. #14556128Q8semidirect
20. #14556125Q8semidirect
127. #2862D71semidirect
132. C71#10362semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225891682
#225868992
#225861982
#225861732
C71×#28622
#1036271